Prove that for any subspace W of Rn W W 0 That is prove tha

Prove that for any subspace W of R^n. W W = {0}. That is, prove that if x is in W and x is in then x must be 0.

Solution

If vectors v1, . . . , vp span a subspace W and if x is orthogonal to each v j for j = 1, . . . , p then x is in W . TRUE since any vector in W can be written as linear combination of basis vectors and dot product splits up nicely over sums and constants.

W= {v1....vp}

then W\' = {u1,u2....up}

so if x belongs to W then it won\'t be in W\'

so W ^ W\' ={0}